Theorem rmo3 3134

Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	|- F/yph

Assertion

Ref	Expression
rmo3	|- (E*x e. A ph <-> A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y))

Distinct variable group:   x,y,A

Allowed substitution hints:   ph(x,y)

Proof of Theorem rmo3

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
2		sban 2069	. . . . . . . . . . 11 |- ([y / x](x e. A /\ ph) <-> ([y / x]x e. A /\ [y / x]ph))
3		clelsb1 2455	. . . . . . . . . . . 12 |- ([y / x]x e. A <-> y e. A)
4	3	anbi1i 676	. . . . . . . . . . 11 |- (([y / x]x e. A /\ [y / x]ph) <-> (y e. A /\ [y / x]ph))
5	2, 4	bitri 240	. . . . . . . . . 10 |- ([y / x](x e. A /\ ph) <-> (y e. A /\ [y / x]ph))
6	5	anbi2i 675	. . . . . . . . 9 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)))
7		an4 797	. . . . . . . . 9 |- (((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
8		ancom 437	. . . . . . . . . 10 |- ((x e. A /\ y e. A) <-> (y e. A /\ x e. A))
9	8	anbi1i 676	. . . . . . . . 9 |- (((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)) <-> ((y e. A /\ x e. A) /\ (ph /\ [y / x]ph)))
10	6, 7, 9	3bitri 262	. . . . . . . 8 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((y e. A /\ x e. A) /\ (ph /\ [y / x]ph)))
11	10	imbi1i 315	. . . . . . 7 |- ((((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (((y e. A /\ x e. A) /\ (ph /\ [y / x]ph)) -> x = y))
12		impexp 433	. . . . . . 7 |- ((((y e. A /\ x e. A) /\ (ph /\ [y / x]ph)) -> x = y) <-> ((y e. A /\ x e. A) -> ((ph /\ [y / x]ph) -> x = y)))
13		impexp 433	. . . . . . 7 |- (((y e. A /\ x e. A) -> ((ph /\ [y / x]ph) -> x = y)) <-> (y e. A -> (x e. A -> ((ph /\ [y / x]ph) -> x = y))))
14	11, 12, 13	3bitri 262	. . . . . 6 |- ((((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (y e. A -> (x e. A -> ((ph /\ [y / x]ph) -> x = y))))
15	14	albii 1566	. . . . 5 |- (A.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> A.y(y e. A -> (x e. A -> ((ph /\ [y / x]ph) -> x = y))))
16		df-ral 2620	. . . . 5 |- (A.y e. A (x e. A -> ((ph /\ [y / x]ph) -> x = y)) <-> A.y(y e. A -> (x e. A -> ((ph /\ [y / x]ph) -> x = y))))
17		r19.21v 2702	. . . . 5 |- (A.y e. A (x e. A -> ((ph /\ [y / x]ph) -> x = y)) <-> (x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
18	15, 16, 17	3bitr2i 264	. . . 4 |- (A.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
19	18	albii 1566	. . 3 |- (A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
20		nfv 1619	. . . . 5 |- F/y x e. A
21		rmo2.1	. . . . 5 |- F/yph
22	20, 21	nfan 1824	. . . 4 |- F/y(x e. A /\ ph)
23	22	mo3 2235	. . 3 |- (E*x(x e. A /\ ph) <-> A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y))
24		df-ral 2620	. . 3 |- (A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
25	19, 23, 24	3bitr4i 268	. 2 |- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y))
26	1, 25	bitri 240	1 |- (E*x e. A ph <-> A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544  [wsb 1648   e. wcel 1710  E*wmo 2205  A.wral 2615  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rmo 2623

This theorem is referenced by: (None)